Sliding mode control (SMC) is a relatively new development in control systems which was first developed in Russia in the 1950's. It is a subclass of Variable Structure Control (VSC), control systems that switch between structures in a non-linear manner in order to drive phase states of the system toward a phase plane trajectory. The phase states of the system include the position error and its derivatives (velocity, acceleration, etc.); the location of the phase states within a phase plane define the state of the system at any given time, and the movement of the phase states through the phase plane is referred to as the phase state trajectory.
When the SMC is operating in a first structure, the phase states follow a first phase state trajectory, and when the SMC is operating in a second structure the phase states follow a second phase state trajectory. By switching between the first and second structures, the phase states are driven toward a third phase state trajectory, referred to as the sliding line (or hyperplane for higher order systems), defined within the phase plane where the first and second phase state trajectories intersect in opposite directions. The switching action is controlled by the location of the phase states relative to the sliding line; when the phase states cross the sliding line while following the first phase state trajectory, the system switches to the second structure to drive the phase states toward the sliding line by following the second phase state trajectory. In this manner, the phase states continuously switch across the sliding line as they follow the sliding line toward the origin of the phase plane (i.e., sliding mode). SMC has the advantage in that the closed loop response is defined by parameters in the controller and it is substantially insensitive to parameter variations in the plant and external load disturbances.
Initially, SMC systems were developed and implemented in continuous-time wherein the phase states are continuously monitored such that the system switches structures instantaneously when the phase states cross the sliding line. An inherent problem with this approach is that the continuous switching action may induce undesirable noise in the system (electrical and acoustic) and it may excite modelled (as well as unmodelled) system dynamics. The above referenced co-pending patent application entitled "Improved Chatter Reduction in Sliding Mode Control of a Disk Drive Actuator" discloses a method for reducing the amount of switching noise by defining a boundary layer around the sliding line and switching between structures only when the phase states exceed the boundary layer. Although this technique reduces the switching noise, there are other inherent problems with continuous-time SMC. Namely, to achieve the desired robustness to parameter variations and external disturbances, it can require gains in the individual structures that exceed the control effort limitations (e.g., exceed the available drive current).
Discrete-time SMC is a more recent development which addresses the drawbacks of continuous-time SMC by combining a conventional linear control effort with a discrete-time sliding mode control effort. This approach is discussed by Weibing Gao in "Discrete-Time Variable Structure Control Systems," IEEE Transactions on Industrial Electronics, Vol. 42, No. Apr. 2, 1995. With discrete-time SMC, the phase states are monitored in discrete-time and the system switches between structures at the sampling rate rather than continuously. Thus, there is an inherent boundary layer about the sliding line with a width defined by the sampling period, as well as other parameters of the controller. Rather than switch continuously such that the phase states "slide" along the sliding line, the phase states switch across the sliding line in a zigzag manner while sliding toward the origin of the phase plane. Another characteristic of discrete-time SMC is that nominally the phase states will cross the sliding line at every sampling period, and the absolute magnitude of the distance between the phase states and the sliding line will remain within the inherent boundary layer.
The position error phase state is typically generated by subtracting an estimated position from a reference position. In disk storage systems, for example, the position error is generated as the difference between the centerline of a target track and the estimated position of the read head. Although the reference position (the track centerline) is static with respect to the disk, the eccentricities of the disk will induce a perturbation into the reference position with respect to the read head, particularly in optical disk storage devices where the media is removable. This perturbation signal is referred to as "runout" and it is typically dominated by a fundamental sinusoid at a frequency equal to the angular velocity of the disk, and a linear combination of the fundamental harmonics. The discrete-time sliding mode control effort is a function of a psuedo derivative of the reference (PDR) signal; it is necessary to generate an estimate of the PDR signal in order to implement a discrete-time sliding mode controller.